Volume

The formula sheet shows how to calculate the volume of a sphere.

\(7400\text{ cm}^3\)

What is the height of the cylinder itself, without the hemisphere on top?

\(3900\text{ cm}^3\)

Substitute the values you know into the formula provided on the formula sheet.

\(10.0\text{ cm}\)


Equations and Inequations

Be very careful if you find yourself dividing by a negative number. It is usually best to avoid it.

\(3>x\) or \(x<3\)

The bracket in the first line is being multiplied by negative 2.

\(x<\frac{3}{2}\) or \(\frac{3}{2}>x\)

Multiplying through carefully by the denominator will deal with the fraction.

\(x=\frac{13}{2}\)

Multiplying by the denominator is a good strategy even when the denominator contains a letter.

\(x=3\)

Find an expression for the perimeter of each shape first. You are told they have equal perimeter...

\(10\text{ cm}\)


Circle Geometry

Look for right-angles, and isosceles triangles.

\(\angle ABD=61^{\circ}\)

Look for right-angles, and isosceles triangles.

\(\angle PQR = 34^{\circ}\)


Scientific Notation

Make sure you can enter numbers in scientific notation into your calculator to perform calculations.

\(8.72\times10^{10}\text{ miles}\)

Make sure you can enter numbers in scientific notation into your calculator, and consider whether they should be put them in brackets when performing calculators.

\(18\)


Expanding Brackets

Take care with negatives when expanding brackets.

\(6x^2+5x-3\)

Check the six terms obtained from expanding these brackets are correct carefully before you collect like terms.

\(x^3+2x^2-13x+10\)

Take care with multiplying a negative by a negative.

\(2x^3-5x^2-13x+4\)

Tackle the expansion of \((x-3)(x+1)\) then consider the "\(-\)" that lies in front of it.

\(-x+3\)


Similar Shapes

Draw the two triangles separately.

\(x=6\)

Obtain the LSF first, and then think about which scale factor is needed.

\(7500\text{ cm}^3\)


Percentages

"100% + 2.5%"

\(£344.61\)

What does appreciate mean?

\(£148 000\)

What percentage of their original pay is £11.18?

\(£10.75\)

Without a calculator, finding 1% will be awkward. What other percentage could be reached, which would let you then find 100% quickly?

\(32\)


Numerical Fractions

When multiplying (or dividing) factions, the denominators don't need to be the same. However, you don't want either to be a mixed fraction.

\(\dfrac{26}{11}\) or \(2\dfrac{4}{11}\)

How can a division by a fraction be rewritten as a multiplication?

\(\dfrac{20}{9}\) or \(2\dfrac{2}{9}\)

When adding (or subtracting) fractions, the denominators need to be equal.

\(\dfrac{193}{30}\) or \(6\dfrac{13}{30}\)

"\(\dfrac{3}{4}\) of..." is the same as "\(\dfrac{3}{4}\times\)..."

\(\dfrac{15}{8}\) or \(1\dfrac{7}{8}\)


Pythagoras

Isolate a right-angled triangle within the diagram to focus on.

\(6.56\text{ mm}\)

Start by calculating the length of line AC, which lies beneath line AD.

\(8.37\text{ m}\)

Since line JF lies East-West, the question is asking whether line FS makes a right-angle with JF.

No, the Second Buoy is not directly South of the First Buoy (with justification).


Straight Lines

Start by using a formula to find the gradient between the points.

\(y=2x+8\)

Start by using a formula to find the gradient between the points.

\(y=-\frac{3}{2}x+11\) or \(2y=-3x+22\)

First rearrange the equation to make \(y\) the subject: \(y=\dots\)

\((0,2)\)

First rearrange the equation to make \(y\) the subject: \(y=\dots\)

\(m=\frac{3}{2}\)

Start by finding two data points which lie directly on the line of best fit, and write down their coordinates.

Equation: \(W=10H+10\)

Weight: \(250\text{ grams}\)


Functions

Use brackets when substituting values into functions.

\(68\)

Take care with negatives.

\(-2\)

The questions is not asking for 17 to be substituted into the function in place of \(x\).

\(a=7\)

You can always check your answer for \(t\) is correct by substituting into into the function afterwards.

\(t=-\frac{4}{3}\)


Factorising

There are three kinds of factorising to consider in turn: common factor, difference of two squares and trinomials/double brackets. Sometimes more than one kind is needed in a single factorising question.

\(2(x+3)(x-3)\)

\(9m^2\) can be written as \((\dots)^2\)?

\((3m+5)(3m-5)\)

"Think of two numbers that multiple to give... and add to give..."

\((x-4)(x-2)\)

It can help to write out all of the possible pairs of whole numbers that multiply to give -20.

\((y+5)(y-4)\)

Always look to take a common factor first. However, there may still be more to do afterwards...

\(3(x+3)(x-1)\)

One approach is to find a pair of numbers that multiply to give 5, but that add to give -11 when one of them is multiplied by 2 first.

\((2x-1)(x-5)\)

Once you think you have the correct answer, you can always multiply your brackets out again to check you get back to the original expression.

\((3x-1)(x+2)\)


Statistics

Start by putting the numbers in order...

  1. Median = 21, interquartile range = 16.
  2. Compare your comments to those in the solution, or ask your teacher to check them.

There are two formulae on the formula sheet for standard deviation. Check your notes to see which one your teacher taught you how to use.

  1. Mean = 17, standard deviation = 9.7.
  2. Compare your comments to those in the solution, or ask your teacher to check them.


Quadratics

To solve a quadratic equation, you will usually want to factorise first.

\(x=-2,x=5\)

"\(y\)-intercepts when \(x=0\)..." "Roots when \(y=0\)..." "The \(x\)-coordinate of the turning point lies in the middle of the roots... "

\(y\)-intercept at \((0,-15)\) Roots at \((5,0)\) and \((-3,0)\) Minimum turning point at \((1,-16)\)

Make sure you label all your coordinates on the sketch. If any numbers don't make sense, check to see if you have done something wrong.

\(y\)-intercept at \((0,-7)\) Roots at \((1,0)\) and \((-7,0)\) Minimum turning point at \((-3,-16)\)


Arcs and Sectors

An arc is a fraction of a circumference.

\(59.9\text{ cm}\)

This is another example of substituting into a formula then rearranging.

\(60.8^{\circ}\)


Changing the Subject

Deal with \(r\) being within a fraction first.

\(r=\dfrac{3P+k}{2}\)

Deal with \(m\) being within a square root first.

\(m=t^2+a^2\)

As with the previous example, the letter is within a square root. But that doesn't mean you should deal with the square root right away.

\(k=(y-pr)^2\)


Surds

Think of the highest square factor of each number within a root.

\(11\sqrt{3}\)

Take care not to forget the 3 in front of the first surd.

\(19\sqrt{2}\)

Multiply by \(\dfrac{?}{?}\)...

\(\dfrac{3\sqrt{35}}{7}\)

Always check whether the surd can be simplified.

\(\dfrac{2\sqrt{3}}{3}\)

Numerical terms can be collected together, and surds which are alike can be collected together. It isn't always possible to collect things together...

\(2\sqrt{3}-6\)


Indices

Learn your index laws carefully.

\(6x^2\)

Remember that a negative power means division.

\(\dfrac{4}{x^2}\)

The order of operations means apply the power (of a power) before the multiplication (of powers).

\(\dfrac{1}{p^5}\)

Remember that both the \(y\) term and the fraction in front are being squared.

\(\dfrac{9}{16}y^{10}\)


Trig Graphs

One of the letters is the amplitude, the other is the number of waves in 360 degrees.

\(a=5,b=3\)

The graph is only shown up to 180 degrees...

\(a=3,b=2\)

The graphs has been moved up, compared to a \(y=a\sin{x^\circ}\) graph.

\(a=3,b=2\)


Simultaneous Equations

The equations must be scaled first.

\(x=5,y=1\)

Take care with negatives, as usual.

\(p=3,q=-2\)

Communicate your answer fully at the end.

A full price ticket costs £7. A concession ticket costs £5.


Area and Sine Rules

Make your you give a complete answer, including units.

\(11.6\text{ m}^2\)

Remember that the Sine Rule can be written in two different ways. Do you want sides or angles on top?

\(3.7 \text{ cm}\)

Remember that the Sine Rule can be written in two different ways. Do you want sides or angles on top?

\(34.0^\circ\)


Algebraic Fractions

In an SQA exam question, it is likely that the numerator and denominator will have something in common after factorising.

\(\dfrac{x-5}{x-3}\)

To simplify an algebraic fraction like this, you should first factorise.

\(\dfrac{x-5}{3x}\)

To simplify an algebraic fraction like this, you should first factorise.

\(\dfrac{x+10}{x-2}\)